Conducting Reporting

1Glenn Research Center

Conducting and Reporting the Results of

a CFD Simulation

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Objective is to achieve a high level of credibility and confidence in the results from CFD performed as part of the design andanalysis of a propulsion system.

Credibility and confidence are obtained by demonstrating acceptable levels of error and uncertainty as assessed through verification and validation.

This discussion is focused on the question

How good are your CFD results?

Determining this precisely is often difficult, but we can try to understand sources of error and ways to demonstrate quality CFD results.

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1. Provide Qualitative Information.Provide features of flow field (conceptual, rough estimates

of quantities). Accuracy requirements are low.

2. Provide Incremental Quantities, P. P = Pbaseline + P Errors partially cancel for P P = (P2-P1)actual + (E2-E1) = Pactual + E Moderate accuracy requirement.

3. Provide Absolute Quantities, P.Requires high level of accuracy.

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The CFD analysis may provide: Steady-state flow at cruise conditions. Features of the shock system (positions, cowl stand-off). Features of the boundary layers (thickness, separation). Pressure recovery at throat and compressor face. Spillage (side walls, cowl). Bleed requirements (slots, bleed holes). Distortion at compressor face. Drag. Off-design flow features. Unsteady flow features (unstart, stability). Aerodynamic and thermal loads.

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Several aspects with regard to conducting and reporting the results of a CFD simulation: Uncertainty and Error Verification and Validation Iterative convergence Grid convergence When things go wrong

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UncertaintyA potential deficiency in any phase or activity of the modeling process

that is due to the lack of knowledge. (AIAA-G-077-1998).The use of potential indicates that deficiencies may or may not exist.

ErrorA recognizable deficiency in any phase or activity and simulation that is

not due to lack of knowledge. (AIAA-G-077-1998).The deficiencies are knowable up examination. Acknowledged errors have

procedures for identifying and removing them. Unacknowledged errors are undiscovered and no set procedures exist to find them.

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Total Error = Physical Modeling Error+ Discretization Error+ Programming Error+ Computer Round-off Error+ Usage Error+ Post-Processing Error

Physical modeling errors are usually acknowledged and may be quantified. Discretization errors are grid dependent but a priori knowledge of the

proper grid for a desired level of accuracy is generally lacking.Programming errors should have been minimized by programmers. Computer round-off errors are usually negligible or controllable. Usage errors are minimized by proper training.Post-processing errors are usually controllable.

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Many physical models are used in CFD and we must understand the limits and possible uncertainty and errors with each model: Spatial Dimensions. (2D, axisymmetric, 3D) Temporal Dimensions. (steady-state, unsteady) Equations. (Navier-Stokes, approximations) Turbulence Model. Thermodynamic and Transport Property model. Chemistry Model. Flow Boundary Conditions (inflow, outflow). Compressor Face Model. Bleed / Blowing Model. Vortex Generator Model.

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Errors that exist because continuum flow equations and physical models are represented in a discrete domain of space (grid) and time.

Level of error is function of interactions between the solution and the grid. Since solution is unknown before-hand, the grid generation may not produce the optimum grid.

Discretization error is quantified through verificationmethods that examine grid convergence.

Discretization error is controllable.

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Glenn Research Center

VerificationThe process of determining that a model implementation

accurately represents the developers conceptual description of the model and the solution to the model. (AIAA-G-077-1998).

Verification checks that there are no programming errors and that the coding correctly implements the equations and models. It also examines discretization errors in the CFD calculations.

ValidationThe process of determining the degree to which a model is an

accurate representation of the real world from the perspective of the intended uses of the model. (AIAA-G-077-1998).

Validation checks that the CFD results agree with reality as observed in experiments, flight tests, or applications.

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Verification assessment has two aspects:

Verification of a CodeVerifies that code has no programming errors and that the codingcorrectly implements the equations and models. The methods involve examining modules of code, checking basic assumptions (mass conservation), and comparing results to analytic results.

Verification of a CalculationVerifies that a calculation (simulation) demonstrates a certain level of accuracy. The primary method for the verification of a calculation is the grid convergence study.

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Validation assessment addresses if CFD code agrees with reality through comparison to experiments. Checks uncertainty in physical models.

Experiment must also address its own errors. Unit Benchmark Subsystem Complete System. Set of validation cases are being collected and studied that

cover the range of flow features present in inlets (turbulent boundary layers, shocks, shock / boundary layer interactions, subsonic ducts, supersonic diffusers).

NPARC Alliance Validation Archive:www.grc.nasa.gov/www/wind/valid

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Convergence used in two ways:

Iterative ConvergenceSimulations should demonstrate iterative convergence. As the algorithm iterates the solution, the simulation results approach a fixed value (residuals drop and level off). This applies to both steady and unsteady flow solutions. In the case of unsteady flows, the iterative convergence applies to iterations over a time step.

Grid ConvergenceAs the grid spacing is reduced, the simulation results become insensitive to the grid and approach the continuum results.

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Iterative Convergence can be demonstrated in several ways:

1. Value of the largest change in the solution over an iteration, Qbig.2. Iteration history of the L2 norm of the equation residual.

3. Examine conservative variables.

4. Iteration history of a local or global quantity of the flow.

Usually some measure of all of these are used to determine iterative convergence.

The largest change in the solution over an iteration is simply

Iterative Convergence

( )miBig QQ ,max =

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The L2 norm of Q is commonly called the residual of the equation solution and is computed as

where N, total number of grid pointsM, number of elements in Q (5 for Navier-Stokes Equations)

It represents the change in the solution over an iteration averaged over all the grid points and equations.

The L2 residual is usually displayed and plotted as its log values, [log10(L2)]. This shows the order-of-magnitude of the change in the solution. Generally desire the L2 residual to approach zero with iterations.

( )NM

QL

N

i

M

m

mi

*

2 1 1

2

,= =

=

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Order-of-magnitude drop (about 5)

Leveling off of residual

psuedo-unsteadiness

Navier-Stokes Equations

The behavior of the residual is not always so smooth.

Unsteadiness (real or pseudo) in the solution (typical of turbulent and separated flows) may limit how far the residual drops.

Often a couple of orders-of-magnitude drop is all one achieves.

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Spalart-Allmaras turbulence model equations

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The solution of the conservation equations must satisfy those conservative principles. Iterative convergence can be demonstrated through examination of the conservative variables:

For example, one can examine the conservation of mass through a duct:

Variation of the mass flow through the duct.

Statistical analysis of the variation. This indicates the level of iterative convergence error in determining the mass flow through the duct.

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The change in a local or global property is perhaps the best measure of iterative convergence since it typically directly monitors the quantity that is important for the engineering analysis.

Lift on the ONERA M6 wing with number of cycles.

Local quantity is point value. Global quantity is integrated over a portion of the flow field.

Examples include: Lift on a wing Drag on a wing Inlet recovery Heat flux

Global quantities usually show a smooth convergence.

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Quantifies:1. ordered discretization error band (related to grid size by order p)2. p, order of grid convergence (order-of-accuracy)3. continuum or zero grid spacing value of observed quantity

Approach: Assumed or demonstrated that all other error terms are negligible,

minimized, accepted, or under control. Perform CFD solution on two or more grids of increased refinement. Solutions must be in the asymptotic range of convergence. If three solutions, can then compute order of convergence, p. Use Richardson extrapolation to compute continuum value. Compute Grid Convergence Indices (GCI) as discretization error band.

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Viscous wall spacing set for y+ < 1.0 for fine grid. Flow field resolution set to resolve shock system. Grid density to keep grid spacing ratio below 15%. Use ICEM CFD to construct and GRIDGEN to improve. Subset grid to remove some sub-layer grid points for

wall function. Best if grid refinement ratio, r = h2 / h1 = 2.0 h is grid spacing. Minimum, r > 1.1 WIND allows for grid sequencing (r = 2).

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Example grid convergence study with three grids.

Mach 2.35. r = h2 / h1 = 2. Asymptotic range observed

in plot. Richardson extrapolation.

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Order of grid convergence, p. Errors reduce as grid is refined.

h is the grid spacing. C is a constant. Use three values to solve for order of convergence.

For this example, p = 1.786

( ) ... TOHhCfhfE pexact +==

( )rffffp lnln

12

23

=

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Extrapolation of two quantities to continuum value at zero grid spacing.

Provides estimate of error. Extrapolate is of higher order.

For this example, (p2/p0)h=0 = 0.97130

121

10

+= ph

r

ffff

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Standardized method for reporting discretization error. Based on Richardson extrapolation. Error band on fine grid solution. Expressed as percentage.

FS is factor of safety (FS = 1.25 if three or more grids) For this example,

GCI12 = 0.1031% (0.0010) GCI23 = 0.3562% (0.0035)

Pressure recovery is 0.971 with grid error of 0.001.

( )1= psfine rF

GCI

1

12

fff

=

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Solutions are in range in which errors decrease at rate denoted by order of convergence, p.

Check that constant C is indeed constant over solutions,

Check with GCI values over three solutions,

For this example, ratio = 1.002

phEC =

1223 GCIrGCIp

=

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In summary, for a CFD analysis, report the following: Objective of CFD analysis Geometry simplifications Grid resolution Boundary and initial conditions Equations and physical models Algorithm settings Iterative convergence criteria Grid convergence criteria Results (values, plots, visualizations) Errors that can be quantified Sensitivity to models and parameters (turbulence, chemistry)

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A sample of books and resources for learning more on CFD:

Anderson, D.A., Tannehill, J.C., and Pletcher, R.H., Computational Fluid Mechanics and Heat Transfer, McGraw-Hill, 1984.

Anderson, J.D., Computational Fluid Dynamics, McGraw-Hill, 1995.

Hirsch, C., Numerical Computation of Internal and External Flows (2 Volumes), John Wiley & Sons, 1988 & 1990.

Hoffmann, K.A. and S.T. Chiang, Computational Fluid Dynamics (3 Volumes), EES Books, 2000. (also AIAA coarse).

Wilcox, D.C., Turbulence Modeling for CFD, DCW Industries, 1998.

AIAA Courses (CFD, Turbulence)Grid Generation training (ICEM, Pointwise, VGRID)

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Conducting Reporting